Nonnegative Matrices in the Mathematical Sciences || Back Matter

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R ed ist rib ut io n su bje ct to SIA M lic en se or co py rig ht; se e h ttp ://w ww .si am .or g/j ou rna ls/ ojs a.p hp INDEX A Aggregationldisaggregation methods, 312 Algebraic eigenspace, 42 Almost monotone matrix, 120 Assumption of fixed proportion of factor inputs, 249 Asymptotic race of convergence, 172 B Birkhoff theorem on doubly stochastic matrices, 50 on K-nonnegative matrices, 6 Bounds for eigenvalues not equal to the spectral radius, 51, 59 C Chain of classes, 42 length of, 42 Circuit, 34 Class, 39 basic, 39 final, 39 height of, 39 nonbasic, 39 Cogredience, 27 Collatz-Wielandt function, 60 Communication between classes, 39 between indices, 39 Comparision matrix, 142 Complementary pivot algorithm, 295-296 Completely decomposable matrix, 75 Completely monotonic functions, 142-146 Completely positive matrix, 20, 304-306 Completely reducible matrix, 53 Cone, 2, 21 complementary, 276 convex,2 dual, 1 ice cream, 2 indecomposable, 20 pointed, 2 polyhedral, 2 proper, 3, 181 reproducing, 2 simplicial, 3 solid, 2 Cone independence, 71 Convergent matrix, 9 Convex body, 5 Copositive matrix, 20, 295 strictly, 295 with respect to a cone, 23 Copositive plus matrix, 295 cp-rank. 304 Cross-positive matrix, 21 strongly, 21 Cyclic matrix, 34, 309-310 D D-monotone, 119 DAD theorems, 58-59 Degree of an eigenvalue, 6 of a nonnegative matrix, 42 Diagonal of a matrix, 49 Diagonally dominant matrix, 20 generalized column, 186 irreducibly, 186 lower semistrictly, 137 strictly, 137, 186 Directed graph, 29 337 D ow nl oa de d 09 /1 3/ 13 to 1 8. 7. 29 .2 40 . R ed ist rib ut io n su bje ct to SIA M lic en se or co py rig ht; se e h ttp ://w ww .si am .or g/j ou rna ls/ ojs a.p hp 338 Index strongly connected, 30 Distribution vector initial, 213 initial probability, 212 probability, 212 stationary, 213 Doubly nonnegative matrix, 304-306 Doubly stochastic matrix, 48. 66, 240 pattern, 55 Doubly substochastic matrix, 104 Drazin inverse, 118, 147, 198, 227 E Eigenset, 191 Eigenvector, generalized, 42 principal, 7 Equimodular matrices, 142 Ergodic state, 219 Essentially nonpositive matrices, 252, see also Matrices, of Leontief type; Z-matrices Extreme points of matrix polytopes, 100-105 polytope of doubly stochastic matrices. see Birkhoff theorem Face, 3 Factorizable matrix, 75 Finite homogeneous Markov chain, see Markov chain Fully indecomposable matrix, 56 Fundamental matrix associated with a Markov chain, 234 G Gauss-Seidel method, 171 block, 174 point, 174 Generalized left inverse, 118 Generalized right inverse, 118 Generalized ultrametric matrices, 307 Generalized left inverse positive matrix, 121 Green's relations, 64 on the semigroup of nonnegative matrices, 72-73 Group inverse, 118, 226-240 Group monotone matrix, 119 H H-matrix, 185 Hadamard-Fischer inequality, 57-160 Hadamard product, 159 Hawkins-Simon condition, 161, 253 Heat equation, 167 Hitchcock's transportation problem, 110 Idempotent nonnegalive matrices, 65-67 Incidence matrix, 75 Indecomposable matrix, see Irreducible, matrix Index of cyclicity, 34 of primitivity, 46 bounds for, 46-48 of a square matrix, 61, 118, 198 Input matrix, 251-257, 260-266 Input-output table, 247 Inverse eigenvalue problem, 61, 87 for primitive matrices, 302-303 for symmetrie nonnegative matrices, 88-92, 301, 303 with prescribed diagonal elements, 93-98 for symmetrie positive matrices, 92-93 Inverse M-matrix problem, 163 Inverse nonnegative matrix, 306-307 Inverse positive matrix, 137 Irreducible matrix, 27, 29-38 M-matrix, 141, 156, 160, 267 monotone matrix, 116 Jacobi method, 170 block, 174 point, 174 K K-inverse-positive matrix, 112 K-irreducible matrix, 11-13, 15 K-monotone matrix, 112 K-nonnegative matrix, 5 K-nonsingular M-matrix, 112 K-positive matrix, 5, 13-14 K-primitive matrix, 16-18 K-reducible matrix, I 1 D ow nl oa de d 09 /1 3/ 13 to 1 8. 7. 29 .2 40 . R ed ist rib ut io n su bje ct to SIA M lic en se or co py rig ht; se e h ttp ://w ww .si am .or g/j ou rna ls/ ojs a.p hp Index 339 K-semipositive matrix, 9, 114 K-singular M-matrix, 114 K-stochastic matrix, 23 Krein-Rutman theorem, 6 L a,- inverse, 117 L-monotone matrix, 119 L-matrix, 296 Least element, 279 Left inverse, 67 Leontief input-output model, 243 closed, 245, 258-265 feasibility of, 260 open, 245, 251-258 feasibility of, 252 profitability of, 252 Leontief matrix, 282, 296 pre, 282 totally, 282 Lexico-positive vector, 274 Limiting matrix of a Markov chain, 227, 234 Linear complementarity problem, 271 parametric, 281 with upper bounds, 281 over cones, 298 Linear equations, 165 Linear least-squares problem, 178 LP solvability, 285-290 LU decomposition of an M-matrix, 157 LU nonnegative factorization, 86 Lyapunov theorem, 162 M µ-positive matrix, 22 M-matrix, 133 nonsingular, 133-142, 158-164, 166, 181-190,245,253-255, 267-270,272,290-292,294 with property c, 152-156. 200-202, 226-228, 261, 263 singular, 147-164,207,213,242. 310-311 Markov chain, 48, 212-243 absorbing, 221 ergodic, 220 period of, 221 periodic, 220 regular, 220 solutions of using iterative methods, 307-312 Matrices of Leontief type, 252 Maximal subgroups of nonnegative matrices, 71 Monomial matrix, 67 Monotone matrix, 137 in a set, 119 Moore-Penrose generalized inverse, 83, 86,119-127,207.240 N Nearly completely decomposable matrix, 312 Nonlinear complementarity problem, 298 Nonnegative matrix, 26-62 doubly, 304-306 inverse, 306-307 on a set, 119 Nonnegative orthant, 1 Nonnegative rank, 304 Nonnegative rank factorization, 67, 84 Nonnegative stability, 150 Normal form of a cyclic matrix, 32 of a reducible matrix, 43 rL o>-matrix, 160 Open sector, 244 Optimum SOR relaxation parameter, 174 Orthostochastic matrix, 85 Oscillatory matrix, 57 Ostrowski-Reich theorem, 195 Output equilibrium vector, 261 P P-matrix, 134, 272-276 Partly decomposable matrix, 75 Permanent, 51 Perron-Frobenius theorem, 27, 32 in a game of numbers, 312-314 in nonnegative linear systems, 314-315 Perron root, 299-301 Poisson's equation, 196 Polyhedron, 49 Polytope, 49 Positive definite matrix, 20 Positive eigenvector of an irreducible matrix, 37, 60 Positive matrix, 26 Positive semidefinite matrix, 20 Positive stable matrix, 113, 135 D ow nl oa de d 09 /1 3/ 13 to 1 8. 7. 29 .2 40 . R ed ist rib ut io n su bje ct to SIA M lic en se or co py rig ht; se e h ttp ://w ww .si am .or g/j ou rna ls/ ojs a.p hp Index Price vector, 252 Prime matrix, 75-82 Primitive matrix, 28, 45-54, 223-226 Principal minors, 34-35, 149-150 Principal pivot transform, 273 Principal pivoting method, 271-272 Proper splitting, 207 Q-matrix, 276-279 Quadratic form sign equivalent hermitian matrices, 192 R r-monotone matrix, 67 Random walk, 217, 236-241 Rank factorization, 67 Rectangular monotone matrix, 119 Reduced triangular block form, 262, see also Normal form Reducible matrix, 27, 38-45 Regular D-class, 64, 70 Regular element, 64 Regular matrix, 277 Regular progression hypothesis, 291 Regular splitsing, 130 convergent, 138 convergent weak, 138 theorem, 183 weak, 151 Restricted basis simplex method, 291 Right inverse, 67 Safety factor, 291 Schur complement, 159, 293 Schur's determinantal formula, 293 Semiconvergent matrix, 152, 197, 228-229, 235-237 Semigroup, 64 of doubly stochastic matrices, 82-83 of nonnegative matrices, 67--82 Semi-inverse, 64-117 Semimonotone matrix, 119 strictly, 277 Semipositive matrix, 136 Set monotonicity, 127-128 Simple odd cactus, 101 Simple path, 34 length of, 34 Singular linear system, 196-203 SOR method, 169, 172-181 block, 174 p-cyclic, 307-312 point, 174 symmetric, 204 Spectral radius, 6, 31, 52, 53, 56, 159 bounds for, 37, 38, 60 Star-shaped set, 291 Stationary probability distribution vector. 2 12-24 1 Stieltjes matrix, 109, 141, 201, 221-224, 265 Stein-Rosenberg theorem, 187, 201 Stochastic matrix, 48-52, 54, 212, 314 Stoper-Samuelson condition, 270 theorem, 270 Strictly ultrametric matrices, 307 Subpermutation matrix, 104 Successive overrelaxation method, see SOR method T t-matrix, 160, 164 T-monotone matrix, 120 Taussky's unification problem, 164 Term rank, 110 symmetric, 110 Totally nonnegative matrix, 57, 163 Totally positive matrix, 57 Transient state, 219 Transition matrix, 212 standard form, 224 Triangular block form, 39, 54, see also Normal form v Value added vector, 252 van der Waerden conjecture, 51, 62 von Neumann model. 269 W Weak monotone, 120 Z Z-matrix, 279D ow nl oa de d 09 /1 3/ 13 to 1 8. 7. 29 .2 40 . R ed ist rib ut io n su bje ct to SIA M lic en se or co py rig ht; se e h ttp ://w ww .si am .or g/j ou rna ls/ ojs a.p hp